noncompactness

Noncompactness is the property of a set or space that is not compact. In topology, a subset A of a space X is compact if every open cover of A has a finite subcover. In metric spaces, compactness is also equivalent to sequential compactness, and in Euclidean spaces to being closed and bounded (the Heine–Borel theorem). Noncompactness then means that one of these compactness conditions fails.

In metric spaces, a subset is noncompact if it is not complete, not totally bounded, or both. For example, the real line R is noncompact; the open interval (0,1) is noncompact even though it is bounded; the closed interval [0,1] is compact. A set can be noncompact in a given ambient space even if its closure in that space is compact, depending on the topology.

Measures of noncompactness are quantitative tools used to describe how far a bounded set is from being compact. The Kuratowski and Hausdorff measures of noncompactness assign a nonnegative number to a bounded set, which is zero exactly when the set is relatively compact. These notions have applications in functional analysis, especially in fixed-point theory and the study of differential and integral equations.

Noncompactness also arises in operator theory: an operator is noncompact if it does not map bounded sets into relatively compact sets. In infinite-dimensional spaces, many natural operators are noncompact, influencing spectral properties and asymptotic behavior.

See also: compactness, Heine–Borel theorem, Arzelà–Ascoli theorem, measures of noncompactness, relative compactness.